332 THE PRINCIPLES OF SCIENCE. 



a moving slip of paper, so that equal intervals of time are 

 represented by equal lengths. There is, perhaps, a ten 

 dency to reduce all comparisons to the comparison of space 

 magnitudes, but in any case one of the senses must be the 

 ultimate judge of coincidence or non-coincidence. 



Since the equation to be established may exist between 

 any multiples or submultiples of the quantities compared, 

 there naturally arise several different modes of comparison 

 adapted to different cases. Let p be the magnitude to 

 be measured, and q that in terms of which it is to be 

 expressed. Then we wish to find such numbers x and y, 



that the equation p = ~q may be true. Now this same 



equation may be presented in four slightly different forms, 

 namely : 



First Form. Second Form. Third Form. Fourth Form. 

 x y PI 



P = - q p~ = q py = qx - - - 



y x x y 



Each of these modes of expressing the same equation cor 

 responds to one mode of effecting a measurement. 



When the standard quantity is greater than that to be 

 measured, we often adopt the first mode, and subdivide 

 the unit until we get a magnitude equal to that measured. 

 The angles observed in surveying, in astronomy, or in 

 goniometry are usually smaller than a whole revolution, 

 and the measuring circle is divided by the use of the 

 microscope and screw, until we obtain an angle undistin- 

 guishable from that observed. The dimensions of minute 

 objects are determined by subdividing the inch or centi 

 metre, the screw micrometer being the most accurate 

 means of subdivision. Ordinary temperatures are esti 

 mated by division, of the standard interval between the 

 freezing and boiling points of water, as marked on a 

 thermometer tube. 



In a still greater number of cases, perhaps, we multiply 



